The GEF estimate builds upon the numerical solution of a dynamical system of equations, while the equilibrium and maximal estimates follow from simple algebraic arguments.
We therefore expect that the GEF estimate is also suitable for more dynamical situations, while the equilibrium and maximal estimates only become relevant after a sufficiently long equilibration time, such that all quantities of interest have reached their time-independent attractor values.
Since the results in Refs. [1]}, [2]} were derived in the electric picture, the fermion current in Eq. (REF ) is treated
as a vector that is primarily controlled by the electric field, \(\mathbf {J} = J/E\,\mathbf {E}\) .
Here, the ratio \(g_YJ/E\)  plays the role of a generalized comoving conductivity for the fermion gas generated during axion inflation,
\(\sigma = \frac{g_YJ}{E} = \frac{41g_Y^3}{72\pi ^2} \frac{B}{a H} \coth \left(\pi \frac{B}{E}\right) \,.\) 
